dc.contributor.author |
Phan, Christopher Lee, 1980- |
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dc.date.accessioned |
2010-05-15T00:13:21Z |
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dc.date.available |
2010-05-15T00:13:21Z |
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dc.date.issued |
2009-06 |
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dc.identifier.uri |
http://hdl.handle.net/1794/10367 |
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dc.description |
xi, 95 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. |
en_US |
dc.description.abstract |
We investigate some homological properties of graded algebras. If A is an R -algebra, then E (A) := Ext A ( R, R ) is an R-algebra under the cup product and is called the Yoneda algebra. (In most cases, we assume R is a field.) A well-known and widely-studied condition on E(A) is the Koszul property. We study a class of deformations of Koszul algebras that arises from the study of equivariant cohomology and algebraic groups and show that under certain circumstances these deformations are Poincaré-Birkhoff-Witt deformations.
Some of our results involve the [Special characters omitted] property, recently introduced by Cassidy and Shelton, which is a generalization of the Koszul property. While a Koszul algebra must be quadratic, a [Special characters omitted] algebra may have its ideal of relations generated in different degrees. We study the structure of the Yoneda algebra corresponding to a monomial [Special characters omitted.] algebra and provide an example of a monomial [Special characters omitted] algebra whose Yoneda algebra is not also [Special characters omitted]. This example illustrates the difficulty of finding a [Special characters omitted] analogue of the classical theory of Koszul duality.
It is well-known that Poincaré-Birkhoff-Witt algebras are Koszul. We find a [Special characters omitted] analogue of this theory. If V is a finite-dimensional vector space with an ordered basis, and A := [Special characters omitted] (V)/I is a connected-graded algebra, we can place a filtration F on A as well as E (A). We show there is a bigraded algebra embedding Λ: gr F E (A) [Special characters omitted] E (gr F A ). If I has a Gröbner basis meeting certain conditions and gr F A is [Special characters omitted], then Λ can be used to show that A is also [Special characters omitted].
This dissertation contains both previously published and co-authored materials. |
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dc.description.sponsorship |
Committee in charge: Brad Shelton, Chairperson, Mathematics;
Victor Ostrik, Member, Mathematics;
Christopher Phillips, Member, Mathematics;
Sergey Yuzvinsky, Member, Mathematics;
Van Kolpin, Outside Member, Economics |
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dc.language.iso |
en_US |
en_US |
dc.publisher |
University of Oregon |
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dc.relation.ispartofseries |
University of Oregon theses, Dept. of Mathematics, Ph. D., 2009; |
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dc.subject |
Koszul properties |
en_US |
dc.subject |
Noncommutative graded algebras |
en_US |
dc.subject |
Yoneda algebra |
en_US |
dc.subject |
Grobner bases |
en_US |
dc.subject |
Homological algebra |
en_US |
dc.subject |
Mathematics |
en_US |
dc.subject |
Algebra, Homological |
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dc.subject |
Algebra, Yoneda |
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dc.subject |
Koszul algebras |
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dc.title |
Koszul and generalized Koszul properties for noncommutative graded algebras |
en_US |
dc.type |
Thesis |
en_US |